Galois orbit of Artin representations 4.3e2_7e2_37e3.8t21.4

• Label: 4.3e2_7e2_37e3.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 3^{2} \cdot 7^{2} \cdot 37^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$22337973= 3^{2} \cdot 7^{2} \cdot 37^{3} $
Artin number field:	Splitting field of $f= x^{8} - 6 x^{6} - 12 x^{5} + 23 x^{4} + 78 x^{3} + 3 x^{2} - 411 x + 367 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 211 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 6 + 129\cdot 211 + 73\cdot 211^{2} + 35\cdot 211^{3} + 130\cdot 211^{4} + 167\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 28 + 10\cdot 211 + 117\cdot 211^{2} + 162\cdot 211^{3} + 102\cdot 211^{4} + 149\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 61 + 28\cdot 211 + 106\cdot 211^{2} + 148\cdot 211^{3} + 158\cdot 211^{4} + 10\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 128 + 9\cdot 211 + 162\cdot 211^{2} + 200\cdot 211^{3} + 115\cdot 211^{4} + 36\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 130 + 31\cdot 211 + 159\cdot 211^{2} + 116\cdot 211^{3} + 61\cdot 211^{4} + 36\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 145 + 203\cdot 211 + 3\cdot 211^{2} + 19\cdot 211^{3} + 144\cdot 211^{4} + 95\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 149 + 57\cdot 211 + 170\cdot 211^{2} + 68\cdot 211^{3} + 37\cdot 211^{4} + 54\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 197 + 162\cdot 211 + 51\cdot 211^{2} + 92\cdot 211^{3} + 93\cdot 211^{4} + 82\cdot 211^{5} +O\left(211^{ 6 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(1,3)(2,6)$

$(1,8,3,7)(2,4,6,5)$

$(1,6)(2,3)(4,7)(5,8)$

$(1,2,3,6)(7,8)$

$(1,3)(2,6)(4,5)(7,8)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,3)(2,6)(4,5)(7,8)$	$-4$
$2$	$2$	$(1,6)(2,3)(4,7)(5,8)$	$0$
$2$	$2$	$(1,2)(3,6)(4,7)(5,8)$	$0$
$2$	$2$	$(1,3)(2,6)$	$0$
$4$	$2$	$(1,4)(2,8)(3,5)(6,7)$	$0$
$4$	$4$	$(1,8,3,7)(2,4,6,5)$	$0$
$4$	$4$	$(1,7,2,4)(3,8,6,5)$	$0$
$4$	$4$	$(1,4,2,7)(3,5,6,8)$	$0$
$4$	$4$	$(1,2,3,6)(7,8)$	$0$
$4$	$4$	$(1,6,3,2)(7,8)$	$0$

The blue line marks the conjugacy class containing complex conjugation.